Generalized compositions with a fixed number of parts

TL;DR

This paper explores compositions of integers with a fixed number of parts across multiple types, revealing new relationships among binomial coefficients, Catalan numbers, and Catalan triangle numbers.

Contribution

It introduces generalized compositions with multiple types of parts, establishing novel connections among classical combinatorial sequences.

Findings

Derived new formulas linking compositions to binomial coefficients

Established relationships between compositions and Catalan numbers

Connected compositions to Catalan triangle numbers

Abstract

We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of the Catalan triangle.